Three dimensional space is said to be spherically symmetric if it admits SO(3) as the group of isometries. Under this symmetry condition, the Einstein’s Field equations for vacuum, yields the Schwarzschild Metric as ...Three dimensional space is said to be spherically symmetric if it admits SO(3) as the group of isometries. Under this symmetry condition, the Einstein’s Field equations for vacuum, yields the Schwarzschild Metric as the unique solution, which essentially is the statement of the well known Birkhoff’s Theorem. Geometrically speaking this theorem claims that the pseudo-Riemanian space-times provide more isometries than expected from the original metric holonomy/ansatz. In this paper we use the method of Lie Symmetry Analysis to analyze the Einstein’s Vacuum Field Equations so as to obtain the Symmetry Generators of the corresponding Differential Equation. Additionally, applying the Noether Point Symmetry method we have obtained the conserved quantities corresponding to the generators of the Schwarzschild Lagrangian and paving way to reformulate the Birkhoff’s Theorem from a different approach.展开更多
We investigate the quantum numbers of the pentaquark states Pc+,which are composed of 4(three flavors)quarks and an antiquark,by analyzing their inherent nodal structure in this paper.Assuming that the four quarks for...We investigate the quantum numbers of the pentaquark states Pc+,which are composed of 4(three flavors)quarks and an antiquark,by analyzing their inherent nodal structure in this paper.Assuming that the four quarks form a tetrahedron or a square,and the antiquark is at the ground state,we determine the nodeless structure of the states with orbital angular moment L≤3,and in turn,the accessible low-lying states.Since the inherent nodal structure depends only on the inherent geometric symmetry,we propose the quantum numbers JPof the low-lying pentaquark states P_c^+may be■,■,■and■,independent of dynamical models.展开更多
In this work,we study a generalized double dispersion Boussinesq equation that plays a significant role in fluid mechanics,scientific fields,and ocean engineering.This equation will be reduced to the Korteweg-de Vries...In this work,we study a generalized double dispersion Boussinesq equation that plays a significant role in fluid mechanics,scientific fields,and ocean engineering.This equation will be reduced to the Korteweg-de Vries equation via using the perturbation analysis.We derive the corresponding vectors,symmetry reduction and explicit solutions for this equation.We readily obtain Bäcklund transformation associated with truncated Painlevéexpansion.We also examine the related conservation laws of this equation via using the multiplier method.Moreover,we investigate the reciprocal Bäcklund transformations of the derived conservation laws for the first time.展开更多
The SERS spectrum of C_(60) was obtained by using silver mirror. From the SERS spectrum of C_(60) and the IR spectrum of C_(60), we proposed that C_(60) adsorbed on the silver mirror adopts a more ellipsoidal shape.
Objective To discuss the etiology,course of diseases and prognosis of symmetry peripheral entrapment enuropathies of upper extremity. Methods The etiology, clinical scale and resluts of 14 cases were analyzed betwen 1...Objective To discuss the etiology,course of diseases and prognosis of symmetry peripheral entrapment enuropathies of upper extremity. Methods The etiology, clinical scale and resluts of 14 cases were analyzed betwen 1999 and April 2002. Results In the bilateral tunnel syndrome, the excellent and good rate was 60% at the original side and 80% at the contralateral side occurred later. In the bilateral carpal tunnel syndrome, the excellent and good rate was 67% at the original side and 89% at the contralateral side occurred later. Conclusion The main etiological factor of the bilateral cubital tunnel syndrome was the elbow eversion deformity. The lesions of synovium contributed mainly to the symmetry carpal tunnel syndrome. The symmetry peripheral entrapment neuropathies of upper extremity should be operated on as soon as possible when diagnosis had been made for enhancement of treatment outcome. 5 refs,2 tabs.展开更多
As one of the paradigmatic models of non-equilibrium systems, the asymmetric simple exclusion process(ASEP) has been widely used to study many physical, chemical, and biological systems. The ASEP shows a range of no...As one of the paradigmatic models of non-equilibrium systems, the asymmetric simple exclusion process(ASEP) has been widely used to study many physical, chemical, and biological systems. The ASEP shows a range of nontrivial macroscopic phenomena, among which, the spontaneous symmetry breaking has gained a great deal of attention. Nevertheless,as a basic problem, it has been controversial whether there exist one or two symmetry-broken phases in the ASEP. Based on the mean field analysis and current minimization principle, this paper demonstrates that one of the broken-symmetry phases does not exist in a bidirectional two-lane ASEP with narrow entrances. Moreover, an exponential decay feature is observed,which has been used to predict the phase boundary in the thermodynamic limit. Our findings might be generalized to other ASEP models and thus deepen the understanding of the spontaneous symmetry breaking in non-equilibrium systems.展开更多
This paper is devoted to the investigation of the Landau–Ginzburg–Higgs equation(LGHe),which serves as a mathematical model to understand phenomena such as superconductivity and cyclotron waves.The LGHe finds applic...This paper is devoted to the investigation of the Landau–Ginzburg–Higgs equation(LGHe),which serves as a mathematical model to understand phenomena such as superconductivity and cyclotron waves.The LGHe finds applications in various scientific fields,including fluid dynamics,plasma physics,biological systems,and electricity-electronics.The study adopts Lie symmetry analysis as the primary framework for exploration.This analysis involves the identification of Lie point symmetries that are admitted by the differential equation.By leveraging these Lie point symmetries,symmetry reductions are performed,leading to the discovery of group invariant solutions.To obtain explicit solutions,several mathematical methods are applied,including Kudryashov's method,the extended Jacobi elliptic function expansion method,the power series method,and the simplest equation method.These methods yield solutions characterized by exponential,hyperbolic,and elliptic functions.The obtained solutions are visually represented through 3D,2D,and density plots,which effectively illustrate the nature of the solutions.These plots depict various patterns,such as kink-shaped,singular kink-shaped,bell-shaped,and periodic solutions.Finally,the paper employs the multiplier method and the conservation theorem introduced by Ibragimov to derive conserved vectors.These conserved vectors play a crucial role in the study of physical quantities,such as the conservation of energy and momentum,and contribute to the understanding of the underlying physics of the system.展开更多
The unsteady magnetohydrodynamic flow of an electrically conducting viscous incompressible third grade fluid bounded by an infinite porous plate is studied with the Hall effect. An external uniform magnetic field is a...The unsteady magnetohydrodynamic flow of an electrically conducting viscous incompressible third grade fluid bounded by an infinite porous plate is studied with the Hall effect. An external uniform magnetic field is applied perpendicular to the plate and the fluid motion is subjected to a uniform suction and injection. Similarity transformations are employed to reduce the non-linear equations governing the flow under discussion to two ordinary differential equations (with and without dispersion terms). Using the finite difference scheme, numerical solutions represented by graphs with reference to the various involved parameters of interest are discussed and appropriate conclusions are drawn.展开更多
We investigate the Einstein field equations corresponding to the Weyl-Lewis-Papapetrou form for an axisymmetric rotating field by using the classical symmetry method. Using the invafiance group properties of the gover...We investigate the Einstein field equations corresponding to the Weyl-Lewis-Papapetrou form for an axisymmetric rotating field by using the classical symmetry method. Using the invafiance group properties of the governing system of partial differential equations (PDEs) and admitting a Lie group of point transformations with commuting infinitesimal generators, we obtain exact solutions to the system of PDEs describing the Einstein field equations. Some appropriate canonical variables are characterized that transform the equations at hand to an equivalent system of ordinary differential equations and some physically important analytic solutions of field equations are constructed. Also, the class of axially symmetric solutions of Einstein field equations including the Papapetrou solution as a particular case has been found.展开更多
In this manuscript, we first perform a complete Lie symmetry classification for a higher-dimensional shallow water wave equation and then construct the corresponding reduced equations with the obtained Lie symmetries....In this manuscript, we first perform a complete Lie symmetry classification for a higher-dimensional shallow water wave equation and then construct the corresponding reduced equations with the obtained Lie symmetries. Moreover, with the extended <em>F</em>-expansion method, we obtain several new nonlinear wave solutions involving differentiable arbitrary functions, expressed by Jacobi elliptic function, Weierstrass elliptic function, hyperbolic function and trigonometric function.展开更多
Parity–time(PT) symmetry has been demonstrated in the frame of classic optics. Its applications in laser science have resulted in unconventional control and manipulation of resonant modes. PT-symmetric periodic circu...Parity–time(PT) symmetry has been demonstrated in the frame of classic optics. Its applications in laser science have resulted in unconventional control and manipulation of resonant modes. PT-symmetric periodic circular Bragg lasers were previously proposed. Analyses with a transfer-matrix method have shown their superior properties of reduced threshold and enhanced modal discrimination between the radial modes. However, the properties of the azimuthal modes were not analyzed, which restricts further development of circular Bragg lasers. Here, we adopt the coupled-mode theory to design and analyze chirped circular Bragg lasers with radial PT symmetry. The new structures possess more versatile modal control with further enhanced modal discrimination between the azimuthal modes. We also analyze azimuthally modulated circular Bragg lasers with radial PT symmetry, which are shown to achieve even higher modal discrimination.展开更多
In this article,we explore the famous Selkov–Schnakenberg(SS)system of coupled nonlinear partial differential equations(PDEs)for Lie symmetry analysis,self-adjointness,and conservation laws.Moreover,miscellaneous sol...In this article,we explore the famous Selkov–Schnakenberg(SS)system of coupled nonlinear partial differential equations(PDEs)for Lie symmetry analysis,self-adjointness,and conservation laws.Moreover,miscellaneous soliton solutions like dark,bright,periodic,rational,Jacobian elliptic function,Weierstrass elliptic function,and hyperbolic solutions of the SS system will be achieved by a well-known technique called sub-ordinary differential equations.All these results are displayed graphically by 3D,2D,and contour plots.展开更多
In this research article,the(3+1)-dimensional nonlinear Gardner-Kadomtsov-Petviashvili(Gardner-KP)equation which depicts the nonlinear modulation of periodic waves,is analyzed through the Lie group-theoretic technique...In this research article,the(3+1)-dimensional nonlinear Gardner-Kadomtsov-Petviashvili(Gardner-KP)equation which depicts the nonlinear modulation of periodic waves,is analyzed through the Lie group-theoretic technique.Considering the Lie invariance condition,we find the symmetry generators.The pro-posed model yields eight-dimensional Lie algebra.Moreover,an optimal system of sub-algebras is com-puted,and similarity reductions are made.The considered nonlinear partial differential equation is re-duced into ordinary differential equations(ODEs)by utilizing the similarity transformation method(STM),which has the benefit of yielding a large number of accurate traveling wave solutions.These ODEs are further solved to get closed-form solutions of the Gardner-KP equation in some cases,while in other cases,we use the new auxiliary equation method to get its soliton solutions.The evolution profiles of the obtained solutions are examined graphically under the appropriate selection of arbitrary parameters.展开更多
In this paper,we use the symmetry of the Lie group analysis as one of the powerful tools that deals with the wide class of fractional order differential equations in the Riemann–Liouville concept.In this study,first,...In this paper,we use the symmetry of the Lie group analysis as one of the powerful tools that deals with the wide class of fractional order differential equations in the Riemann–Liouville concept.In this study,first,we employ the classical and nonclassical Lie symmetries(LS)to acquire similarity reductions of the nonlinear fractional far field Korteweg–de Vries(KdV)equation,and second,we find the related exact solutions for the derived generators.Finally,according to the LS generators acquired,we construct conservation laws for related classical and nonclassical vector fields of the fractional far field KdV equation.展开更多
As a celebrated nonlinear water wave equation,the Davey–Stewartson equation is widely studied by researchers,especially in the field of mathematical physics.On the basis of the Riemann–Liouville fractional derivativ...As a celebrated nonlinear water wave equation,the Davey–Stewartson equation is widely studied by researchers,especially in the field of mathematical physics.On the basis of the Riemann–Liouville fractional derivative,the time-fractional Davey–Stewartson equation is investigated in this paper.By application of the Lie symmetry analysis approach,the Lie point symmetries and symmetry groups are obtained.At the same time,the similarity reductions are derived.Furthermore,the equation is converted to a system of fractional partial differential equations and a system of fractional ordinary differential equations in the sense of Riemann–Liouville fractional derivative.By virtue of the symmetry corresponding to the scalar transformation,the equation is converted to a system of fractional ordinary differential equations in the sense of Erdélyi–Kober fractional integro-differential operators.By using Noether’s theorem and Ibragimov’s new conservation theorem,the conserved vectors and the conservation laws are derived.Finally,the traveling wave solutions are achieved and plotted.展开更多
By taking advantage of three different computational analytical methods:the Lie symmetry analysis,the generalized Riccati equation mapping approach,and the modified Kudryashov approach,we construct multiple new analyt...By taking advantage of three different computational analytical methods:the Lie symmetry analysis,the generalized Riccati equation mapping approach,and the modified Kudryashov approach,we construct multiple new analytical soliton solutions to the nonlinear convection-diffusion-reaction equation(NCDR)with power-law nonlinearity and density-dependent diffusion.Lie symmetry analysis is one of the pow-erful techniques that reduce the higher-order partial differential equation into an ordinary differential equation by reduction of independent variables.By the Lie group technique,we obtain one-parameter in-variant transformations,determining equations and corresponding vectors for the considered convection-diffusion-reaction equation.By treating the parameters of the governing equation as constants,the ap-plied methods yield a variety of new closed-form solutions,including inverse function solutions,periodic solutions,exponential function solutions,dark solitons,singular solitons,combo bright-singular solitons,and the combine of bright-dark solitons and dark-bright solitons.Moreover,using the Bäcklund transfor-mation of the generalized Riccati equation and modified Kudryashov method,we can construct multiple solitons and other solutions of the considered equation.The obtained new solutions of this work demon-strate that the used approaches are powerful and effective in dealing with nonlinear equations,and that these solutions are required to explain many biological and physical phenomena.Comparing our obtained solutions of this paper with the ones obtained in the literature,we see that our solutions are new and not reported elsewhere.These newly formed soliton solutions will be more beneficial in the various dis-ciplines of ocean engineering,plasma physics,and nonlinear sciences.展开更多
We investigate and concentrate on new infinitesimal generators of Lie symmetries for an extended(2+1)-dimensional Calogero-Bogoyavlenskii-Schif(eCBS)equation using the commutator table which results in a system of non...We investigate and concentrate on new infinitesimal generators of Lie symmetries for an extended(2+1)-dimensional Calogero-Bogoyavlenskii-Schif(eCBS)equation using the commutator table which results in a system of nonlinear ordinary differential equations(ODEs)which can be manually solved.Through two stages of Lie symmetry reductions,the eCBS equation is reduced to non-solvable nonlinear ODEs using different combinations of optimal Lie vectors.Using the integration method and the Riccati and Bernoulli equation methods,we investigate new analytical solutions to those ODEs.Back substituting to the original variables generates new solutions to the eCBS equation.These results are simulated through three-and two-dimensional plots.展开更多
In this investigation,the(2+1)-dimensions of dispersive long wave equations on shallow waters which are called Wu-Zhang(WZ)equations are studied by using symmetry analysis.The system of partial differential equations ...In this investigation,the(2+1)-dimensions of dispersive long wave equations on shallow waters which are called Wu-Zhang(WZ)equations are studied by using symmetry analysis.The system of partial differential equations are reduced to the type of system of ordinary differential equations.The exact solutions of ordinary differential equations are obtained by the general Kudryashov method[2].Exact solutions including singular wave,kink wave and anti-kink wave are shown.Some figures are given to show the properties of the solutions.展开更多
We solve the Ostrovsky equation in the absence of the rotation effect using the Hirota bilinear method and symbolic calculation.Some unique interaction phenomena have been obtained between lump so-lution,breather wave...We solve the Ostrovsky equation in the absence of the rotation effect using the Hirota bilinear method and symbolic calculation.Some unique interaction phenomena have been obtained between lump so-lution,breather wave,periodic wave,kink soliton,and two-wave solutions.All the obtained solutions are validated by putting them into the original problem using the Wolfram Mathematica 12.The physical characteristics of the solutions have been visually represented to shed additional light on the acquired re-sults.Furthermore,using the novel conservation theory,the conserved vectors of the governing equation have been generated.The discovered results are helpful in understanding particular physical phenomena in fluid dynamics as well as the dynamics of nonlinear higher dimensional wave fields in computational physics and ocean engineering and related disciplines.展开更多
The aim of the present communication is to discuss the analytical solution for the unsteady flow of a third grade fluid which occupies the space y 〉 0 over an infinite porous plate. The flow is generated due to the m...The aim of the present communication is to discuss the analytical solution for the unsteady flow of a third grade fluid which occupies the space y 〉 0 over an infinite porous plate. The flow is generated due to the motion of the plate in its own plane with an impulsive velocity V(t). Translational symmetries in variables t and y are utilized to reduce the governing non-linear partial differential equation into an ordinary differential equation. The reduced problem is then solved using homotopy analysis method(HAM). Graphs representing the solution are plotted and discussed and proper conclusions are drawn.展开更多
文摘Three dimensional space is said to be spherically symmetric if it admits SO(3) as the group of isometries. Under this symmetry condition, the Einstein’s Field equations for vacuum, yields the Schwarzschild Metric as the unique solution, which essentially is the statement of the well known Birkhoff’s Theorem. Geometrically speaking this theorem claims that the pseudo-Riemanian space-times provide more isometries than expected from the original metric holonomy/ansatz. In this paper we use the method of Lie Symmetry Analysis to analyze the Einstein’s Vacuum Field Equations so as to obtain the Symmetry Generators of the corresponding Differential Equation. Additionally, applying the Noether Point Symmetry method we have obtained the conserved quantities corresponding to the generators of the Schwarzschild Lagrangian and paving way to reformulate the Birkhoff’s Theorem from a different approach.
基金supported by the National Natural Science Foundation of China under Contracts No.11435001,No.11775041the National Key Basic Research Program of China under Contract No.2015CB856900。
文摘We investigate the quantum numbers of the pentaquark states Pc+,which are composed of 4(three flavors)quarks and an antiquark,by analyzing their inherent nodal structure in this paper.Assuming that the four quarks form a tetrahedron or a square,and the antiquark is at the ground state,we determine the nodeless structure of the states with orbital angular moment L≤3,and in turn,the accessible low-lying states.Since the inherent nodal structure depends only on the inherent geometric symmetry,we propose the quantum numbers JPof the low-lying pentaquark states P_c^+may be■,■,■and■,independent of dynamical models.
基金supported by Natural Science Foundation of Hebei Province,China(Grant No.A2018207030)Youth Key Program of Hebei University of Economics and Business(2018QZ07)+2 种基金Key Program of Hebei University of Economics and Business(2020ZD11)Youth Team Support Program of Hebei University of Economics and BusinessNational Natural Science Foundation of China(Grant No.11801133)。
文摘In this work,we study a generalized double dispersion Boussinesq equation that plays a significant role in fluid mechanics,scientific fields,and ocean engineering.This equation will be reduced to the Korteweg-de Vries equation via using the perturbation analysis.We derive the corresponding vectors,symmetry reduction and explicit solutions for this equation.We readily obtain Bäcklund transformation associated with truncated Painlevéexpansion.We also examine the related conservation laws of this equation via using the multiplier method.Moreover,we investigate the reciprocal Bäcklund transformations of the derived conservation laws for the first time.
文摘The SERS spectrum of C_(60) was obtained by using silver mirror. From the SERS spectrum of C_(60) and the IR spectrum of C_(60), we proposed that C_(60) adsorbed on the silver mirror adopts a more ellipsoidal shape.
文摘Objective To discuss the etiology,course of diseases and prognosis of symmetry peripheral entrapment enuropathies of upper extremity. Methods The etiology, clinical scale and resluts of 14 cases were analyzed betwen 1999 and April 2002. Results In the bilateral tunnel syndrome, the excellent and good rate was 60% at the original side and 80% at the contralateral side occurred later. In the bilateral carpal tunnel syndrome, the excellent and good rate was 67% at the original side and 89% at the contralateral side occurred later. Conclusion The main etiological factor of the bilateral cubital tunnel syndrome was the elbow eversion deformity. The lesions of synovium contributed mainly to the symmetry carpal tunnel syndrome. The symmetry peripheral entrapment neuropathies of upper extremity should be operated on as soon as possible when diagnosis had been made for enhancement of treatment outcome. 5 refs,2 tabs.
基金Project supported by the National Basic Research Program of China(Grant No.2012CB725404)the National Natural Science Foundation of China(Grant Nos.11422221 and 11672289)
文摘As one of the paradigmatic models of non-equilibrium systems, the asymmetric simple exclusion process(ASEP) has been widely used to study many physical, chemical, and biological systems. The ASEP shows a range of nontrivial macroscopic phenomena, among which, the spontaneous symmetry breaking has gained a great deal of attention. Nevertheless,as a basic problem, it has been controversial whether there exist one or two symmetry-broken phases in the ASEP. Based on the mean field analysis and current minimization principle, this paper demonstrates that one of the broken-symmetry phases does not exist in a bidirectional two-lane ASEP with narrow entrances. Moreover, an exponential decay feature is observed,which has been used to predict the phase boundary in the thermodynamic limit. Our findings might be generalized to other ASEP models and thus deepen the understanding of the spontaneous symmetry breaking in non-equilibrium systems.
基金the South African National Space Agency (SANSA) for funding this work
文摘This paper is devoted to the investigation of the Landau–Ginzburg–Higgs equation(LGHe),which serves as a mathematical model to understand phenomena such as superconductivity and cyclotron waves.The LGHe finds applications in various scientific fields,including fluid dynamics,plasma physics,biological systems,and electricity-electronics.The study adopts Lie symmetry analysis as the primary framework for exploration.This analysis involves the identification of Lie point symmetries that are admitted by the differential equation.By leveraging these Lie point symmetries,symmetry reductions are performed,leading to the discovery of group invariant solutions.To obtain explicit solutions,several mathematical methods are applied,including Kudryashov's method,the extended Jacobi elliptic function expansion method,the power series method,and the simplest equation method.These methods yield solutions characterized by exponential,hyperbolic,and elliptic functions.The obtained solutions are visually represented through 3D,2D,and density plots,which effectively illustrate the nature of the solutions.These plots depict various patterns,such as kink-shaped,singular kink-shaped,bell-shaped,and periodic solutions.Finally,the paper employs the multiplier method and the conservation theorem introduced by Ibragimov to derive conserved vectors.These conserved vectors play a crucial role in the study of physical quantities,such as the conservation of energy and momentum,and contribute to the understanding of the underlying physics of the system.
文摘The unsteady magnetohydrodynamic flow of an electrically conducting viscous incompressible third grade fluid bounded by an infinite porous plate is studied with the Hall effect. An external uniform magnetic field is applied perpendicular to the plate and the fluid motion is subjected to a uniform suction and injection. Similarity transformations are employed to reduce the non-linear equations governing the flow under discussion to two ordinary differential equations (with and without dispersion terms). Using the finite difference scheme, numerical solutions represented by graphs with reference to the various involved parameters of interest are discussed and appropriate conclusions are drawn.
文摘We investigate the Einstein field equations corresponding to the Weyl-Lewis-Papapetrou form for an axisymmetric rotating field by using the classical symmetry method. Using the invafiance group properties of the governing system of partial differential equations (PDEs) and admitting a Lie group of point transformations with commuting infinitesimal generators, we obtain exact solutions to the system of PDEs describing the Einstein field equations. Some appropriate canonical variables are characterized that transform the equations at hand to an equivalent system of ordinary differential equations and some physically important analytic solutions of field equations are constructed. Also, the class of axially symmetric solutions of Einstein field equations including the Papapetrou solution as a particular case has been found.
文摘In this manuscript, we first perform a complete Lie symmetry classification for a higher-dimensional shallow water wave equation and then construct the corresponding reduced equations with the obtained Lie symmetries. Moreover, with the extended <em>F</em>-expansion method, we obtain several new nonlinear wave solutions involving differentiable arbitrary functions, expressed by Jacobi elliptic function, Weierstrass elliptic function, hyperbolic function and trigonometric function.
基金Hong Kong Research Grants Council Early Career Scheme(24208915)National Natural Science Foundation of China(NSFC)Research Grants Council of Hong Kong Joint Research Scheme(N_CUHK415/15)
文摘Parity–time(PT) symmetry has been demonstrated in the frame of classic optics. Its applications in laser science have resulted in unconventional control and manipulation of resonant modes. PT-symmetric periodic circular Bragg lasers were previously proposed. Analyses with a transfer-matrix method have shown their superior properties of reduced threshold and enhanced modal discrimination between the radial modes. However, the properties of the azimuthal modes were not analyzed, which restricts further development of circular Bragg lasers. Here, we adopt the coupled-mode theory to design and analyze chirped circular Bragg lasers with radial PT symmetry. The new structures possess more versatile modal control with further enhanced modal discrimination between the azimuthal modes. We also analyze azimuthally modulated circular Bragg lasers with radial PT symmetry, which are shown to achieve even higher modal discrimination.
文摘In this article,we explore the famous Selkov–Schnakenberg(SS)system of coupled nonlinear partial differential equations(PDEs)for Lie symmetry analysis,self-adjointness,and conservation laws.Moreover,miscellaneous soliton solutions like dark,bright,periodic,rational,Jacobian elliptic function,Weierstrass elliptic function,and hyperbolic solutions of the SS system will be achieved by a well-known technique called sub-ordinary differential equations.All these results are displayed graphically by 3D,2D,and contour plots.
基金The authors would like to thank the Deanship of Scientific Research at Majmaah University for supporting this work under Project R-2022-178.
文摘In this research article,the(3+1)-dimensional nonlinear Gardner-Kadomtsov-Petviashvili(Gardner-KP)equation which depicts the nonlinear modulation of periodic waves,is analyzed through the Lie group-theoretic technique.Considering the Lie invariance condition,we find the symmetry generators.The pro-posed model yields eight-dimensional Lie algebra.Moreover,an optimal system of sub-algebras is com-puted,and similarity reductions are made.The considered nonlinear partial differential equation is re-duced into ordinary differential equations(ODEs)by utilizing the similarity transformation method(STM),which has the benefit of yielding a large number of accurate traveling wave solutions.These ODEs are further solved to get closed-form solutions of the Gardner-KP equation in some cases,while in other cases,we use the new auxiliary equation method to get its soliton solutions.The evolution profiles of the obtained solutions are examined graphically under the appropriate selection of arbitrary parameters.
文摘In this paper,we use the symmetry of the Lie group analysis as one of the powerful tools that deals with the wide class of fractional order differential equations in the Riemann–Liouville concept.In this study,first,we employ the classical and nonclassical Lie symmetries(LS)to acquire similarity reductions of the nonlinear fractional far field Korteweg–de Vries(KdV)equation,and second,we find the related exact solutions for the derived generators.Finally,according to the LS generators acquired,we construct conservation laws for related classical and nonclassical vector fields of the fractional far field KdV equation.
基金the National Natural Science Foundation of China(Grant No.11975143)。
文摘As a celebrated nonlinear water wave equation,the Davey–Stewartson equation is widely studied by researchers,especially in the field of mathematical physics.On the basis of the Riemann–Liouville fractional derivative,the time-fractional Davey–Stewartson equation is investigated in this paper.By application of the Lie symmetry analysis approach,the Lie point symmetries and symmetry groups are obtained.At the same time,the similarity reductions are derived.Furthermore,the equation is converted to a system of fractional partial differential equations and a system of fractional ordinary differential equations in the sense of Riemann–Liouville fractional derivative.By virtue of the symmetry corresponding to the scalar transformation,the equation is converted to a system of fractional ordinary differential equations in the sense of Erdélyi–Kober fractional integro-differential operators.By using Noether’s theorem and Ibragimov’s new conservation theorem,the conserved vectors and the conservation laws are derived.Finally,the traveling wave solutions are achieved and plotted.
文摘By taking advantage of three different computational analytical methods:the Lie symmetry analysis,the generalized Riccati equation mapping approach,and the modified Kudryashov approach,we construct multiple new analytical soliton solutions to the nonlinear convection-diffusion-reaction equation(NCDR)with power-law nonlinearity and density-dependent diffusion.Lie symmetry analysis is one of the pow-erful techniques that reduce the higher-order partial differential equation into an ordinary differential equation by reduction of independent variables.By the Lie group technique,we obtain one-parameter in-variant transformations,determining equations and corresponding vectors for the considered convection-diffusion-reaction equation.By treating the parameters of the governing equation as constants,the ap-plied methods yield a variety of new closed-form solutions,including inverse function solutions,periodic solutions,exponential function solutions,dark solitons,singular solitons,combo bright-singular solitons,and the combine of bright-dark solitons and dark-bright solitons.Moreover,using the Bäcklund transfor-mation of the generalized Riccati equation and modified Kudryashov method,we can construct multiple solitons and other solutions of the considered equation.The obtained new solutions of this work demon-strate that the used approaches are powerful and effective in dealing with nonlinear equations,and that these solutions are required to explain many biological and physical phenomena.Comparing our obtained solutions of this paper with the ones obtained in the literature,we see that our solutions are new and not reported elsewhere.These newly formed soliton solutions will be more beneficial in the various dis-ciplines of ocean engineering,plasma physics,and nonlinear sciences.
文摘We investigate and concentrate on new infinitesimal generators of Lie symmetries for an extended(2+1)-dimensional Calogero-Bogoyavlenskii-Schif(eCBS)equation using the commutator table which results in a system of nonlinear ordinary differential equations(ODEs)which can be manually solved.Through two stages of Lie symmetry reductions,the eCBS equation is reduced to non-solvable nonlinear ODEs using different combinations of optimal Lie vectors.Using the integration method and the Riccati and Bernoulli equation methods,we investigate new analytical solutions to those ODEs.Back substituting to the original variables generates new solutions to the eCBS equation.These results are simulated through three-and two-dimensional plots.
文摘In this investigation,the(2+1)-dimensions of dispersive long wave equations on shallow waters which are called Wu-Zhang(WZ)equations are studied by using symmetry analysis.The system of partial differential equations are reduced to the type of system of ordinary differential equations.The exact solutions of ordinary differential equations are obtained by the general Kudryashov method[2].Exact solutions including singular wave,kink wave and anti-kink wave are shown.Some figures are given to show the properties of the solutions.
文摘We solve the Ostrovsky equation in the absence of the rotation effect using the Hirota bilinear method and symbolic calculation.Some unique interaction phenomena have been obtained between lump so-lution,breather wave,periodic wave,kink soliton,and two-wave solutions.All the obtained solutions are validated by putting them into the original problem using the Wolfram Mathematica 12.The physical characteristics of the solutions have been visually represented to shed additional light on the acquired re-sults.Furthermore,using the novel conservation theory,the conserved vectors of the governing equation have been generated.The discovered results are helpful in understanding particular physical phenomena in fluid dynamics as well as the dynamics of nonlinear higher dimensional wave fields in computational physics and ocean engineering and related disciplines.
基金supported by the National Research Foundation(NRF) of South Africa for research grant
文摘The aim of the present communication is to discuss the analytical solution for the unsteady flow of a third grade fluid which occupies the space y 〉 0 over an infinite porous plate. The flow is generated due to the motion of the plate in its own plane with an impulsive velocity V(t). Translational symmetries in variables t and y are utilized to reduce the governing non-linear partial differential equation into an ordinary differential equation. The reduced problem is then solved using homotopy analysis method(HAM). Graphs representing the solution are plotted and discussed and proper conclusions are drawn.